We state our main result in the framework of a dynamical system given by iteration of a continuous map \(T\)  on a Polish metric space \(X\) . In this setting the sets of interest consist of points whose orbits have prescribed statistical behaviour with respect to a bounded continuous observable \(\varphi \colon X\rightarrow \mathbb {R}\) . Here by “statistical behaviour of an orbit of \(x\in X\) ” we mean the asymptotic behaviour of the sequence of Birkoff averages of \(\varphi \)  along the orbit of \(x\) . In the classical topological dynamics, one usually assumes that the space \(X\)  is compact, but some expansions, most notably, continued fraction expansions come from dynamical systems acting on non-compact Polish spaces, so we work in this greater generality.
We refer the reader to [1]}, [2]}, [3]}, [4]} for related problems for dynamical systems (linear operators, countable Markov chains) on non-compact phase spaces. We study the following questions:
